The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in 2-dimensional space

Hoshiga, Akira

doi:10.14492/hokmj/1249046363

Cited by 13 publications

(12 citation statements)

References 4 publications

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“…Note that analogous results have been obtained in [18] for the nonlinear KleinGordon equations and in [11], [13] for the nonlinear wave equations.…”

Section: Remark 4 When We Putsupporting

confidence: 74%

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

Hayashi

Naumkin²,

Sunagawa

2008

SIAM J. Math. Anal.

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This talk is based on a joint work with Nakao Hayashi and Pavel Naumkin [8]. We consider the initial value problem for the nonlinear Shrödinger equation of the derivative type:whereand u is a complex-valued unknown function. We will occasionally write u x for ∂ x u, and u denotes the complex conjugate of u. The nonlinear term N(u, u x ) is a cubic homogeneous polynomial in (u, u, u x , u x ) with complex coefficients, and it satisfies so-called gauge invariance, that is,The aim of this talk is to present a structural condition on the nonlinear term N under which the corresponding forward Cauchy problem (1) has a disspative nature. To explain the motivation, let us begin with the simplest case where N is independent of u x , i.e., N = λ|u| 2 u with λ ∈ C. Then it is easy to see thatwhich suggests dissipativity if Im λ < 0. In fact, it is proved in [17] that the solution decays like O((t log t)x as t → +∞ if Im λ < 0 and u 0 is small enough. Since the non-trivial free solution (i.e., the solution to (1) for N ≡ 0, u 0 = 0) only decays like O(t −1/2 ), this gain of additional logarithmic time decay reflects a disspative character. Now we turn our attentions to the general gauge-invariant cubic nonlinear terms involving both u and u x . Note that we can not expect the conservation law like (3)

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“…Note that analogous results have been obtained in [18] for the nonlinear KleinGordon equations and in [11], [13] for the nonlinear wave equations.…”

Section: Remark 4 When We Putsupporting

confidence: 74%

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

Hayashi

Naumkin²,

Sunagawa

2008

SIAM J. Math. Anal.

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“…In this paper, we will unify two global existence results mentioned above: One is the global existence result under the quadratic and cubic null conditions (1.4)-(1.5) in [2]; another is the result under the Agemi condition (1.6) and F q (∂u) ≡ 0 in [10] and [21]. We will also investigate a condition, corresponding to (1.8), to ensure that the nonlinearity works as nonlinear dissipation for systems.…”

Section: Introduction and The Main Resultsmentioning

confidence: 93%

Energy decay for systems of semilinear wave equations with dissipative structure in two space dimensions

Katayama

Matsumura

Sunagawa

2014

Nonlinear Differ. Equ. Appl.

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“…The above inequality is the iteration frame for the subcritical case. This inequality is the same as the one in (8,2) in [24], if q is substituted by q. Making use of (7.10) and (7.11), one can obtain Proposition 7.1 immediately by the same argument in [24].…”

Section: [The 2ndmentioning

confidence: 61%

Global existence for semilinear wave equations with the critical blow-up term in high dimensions

Takamura

Wakasa

2016

Journal of Differential Equations

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We are interested in almost global existence cases in the general theory for nonlinear wave equations, which are caused by critical exponents of nonlinear terms. Such situations can be found in only three cases in the theory, cubic terms in two space dimensions, quadratic terms in three space dimesions and quadratic terms including a square of unknown functions itself in four space dimensions. Except for the last case, criterions to classify nonlinear terms into the almost global, or global existence case, are well-studied and known to be so-called null condition and non-positive condition.Our motivation of this work is to find such a kind of the criterion in four space dimensions. In our previous paper, an example of the nonsingle term for the almost global existence case is introduced. In this paper, we show an example of the global existence case. These two examples have nonlinear integral terms which are closely related to derivative loss due to high dimensions. But it may help us to describe the final form of the criterion.

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The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in 2-dimensional space

Cited by 13 publications

References 4 publications

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

Energy decay for systems of semilinear wave equations with dissipative structure in two space dimensions

Global existence for semilinear wave equations with the critical blow-up term in high dimensions

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